CONSTRUCTION OF THE PLAIN SCALE. a a 1. With the radius intended for the scale, describe a semicircle, (see plate 1st fig 1st,) and from the centre C; draw CD, perpendicular to AB, which will divide the semicircle into two quadrants, AD, BD; continue CD towards S, draw BT, perpendicular to CB, and join BD and AD. 2. Divide the quadrant BD into 9 equal parts, then each of these divisions will be 10 degrees. Subdivide each of these parts into single degrees, and if your radius will admit of it, into minutes, or some equal parts of a degree, larger than a minute. 3. Set one foot of the compasses at B, and transfer each of the divisions of the quadrant BD, to the right line BD, then BD will be a Line of Chords. 4. Then the points, 10, 20, 30, &c. in the quadrant BD, draw right lines, parallel to CD, to cut the radios CB, and they will divide that line into a line of sines, which must be numbered from C towards B. 5. If the same line of sines be numbered from B towards C it will become a line of versed sines, which may be continued to 180 degrees, if the same divisions be transferred on the same line on the other side of the centre C. 6. From the centre E, through the several divisions of the quadrant BD, draw right lines till they cut the tangent BT so the line BT, will become a Line of Tangents.* 7. Set one foot of the compasses at C, extend the other to the several divisions, 10, 20, 30, &c. in the tangent line BT, and transfer these extents, severally, to the right line CS, then that line will be a Line of Secants. 8. Right lines draw from A to the several divisions 10, * A mistake was made in engraving this plate. The lines running parallel with C D, and the curved lines between B D, and the lines falling from the Tangents should intersect each other on the semicircle A D B; and the last mentioned lines should run in a dia rection to centre at C. a a a 20, 30, 40, &c. in the quadrant BD will divide the the radius CD into a line of semi-tangents. 9. Divide the quadrant AD into 8 eqnal parts, and from A as a centre, transfer these divisions severally into the line AD, then AD will be a Line of Rhombs, each division answering 11 degrees and 15 minutes upon the Line of Chords. The use of this division, or line, is for measuring angles, and protracting them, according to the common divisions of the Mariner's Compass. If the radius AC be divided into 100, or 1000, &c. equal parts, and the length of the several sines, tangents, and secants, corresponding to the several arches of the quadrant, be measured thereby, and these numbers set down in a table, each in its proper column, you will by these measures have a col. lection of numbers, hy which the several cases in Trigonometry, may be solved. Right lines, graduated as above mentioned, being placed severally upon the Rule, form the instrument called the Plain Scale, (see plate 1st fig 2d,) by which the line and angles of all triangles may be measured. All right lines, as the sides of plain triangles, &c. when they are considered simply as such, without having relation to a circle, are measured by scales of equal parts, each of which is subdivided into ten equal parts ; and this serves as the common divisions to all the rest. In most Scales, an inch is taken for a common measure, and whatever an inch is divided into, may be found at the end of the Scale : divided in this manner, any number less than 100, may be readily taken. But if the number should consist of three places of figures, the value of the third figure, cannot be exactly ascertained ; and in this case, it is better to use a Diagonal Scale, by which any number consisting of three places of figures, may be exactly found. The figures of this Scale are given in plate 1st, fig 3d ; its construction is as follows: Having prepared a ruler of a convenient breadth for your Scale, draw near the edges, two right lines, AF, CC, parallel to each other; divide one of these lines, as FA into equal parts, according to the size of your Scale, and through each of these divisions, draw right lines perpendicular to AF, to meet CC, then divide the breadtla into ten equal parts; and through each of these divisions, draw right lines, parallel to AF and CC. Divide the lines AB, CD, into ten equal parts, and from the point A to the : first division in the line CD, draw a right line parallel to that line ; draw right lines through all their divisions, and the Scale is finished. Besides the lines already mentioned, there is another on some Scales, and on the Gunter's Scale, marked ML, which is joined to a line of chords, and shows how many miles of Easting and Westing corresponds to a degree of Longitude in every degree of Latitude.* These several lines are generally put on one side of a Rule two feet long, and on the other side is laid down a scale of Logarithms of the sine tangents and numbers, which is commonly called Gunter's Scale, and as it is of general use, it requires a particular description, which will be found on the thir a a teenth page. * As it would confuse the adjoined figure, to describe on it the line of Longitude, it is neglected; but the ocnstruction is as follows: divide the line CB, into 60 equal parts, and through each point draw lines parallel to CD to intersect the arch BD Take B as a centre, transfer the several points of intersection, to the line BD, and their number is from D towards B, from 0 to 60, and it will be the Line of Longitude. HOW TO PROVE THE SLIDING RULE. Rule.-Draw out the slider to the right hand, till 1 on the slider coincides with 2 on the fixed part, then 2 on the slider will coincide with 4 on the fixed part. Continue to draw the slider till 1 on the slider coincides with 3 on the fixed part, then 2 on the slider will coincide with 6 on the faxed part; till 1 on the slider coincides with 4 on the fixed part, then 2 on the slider will coincide with 8 on the fixed part; till 1 coincides with 5, then 2 will coincide with the centre 1; till 1 coincides with 5, then 2 will coincide with 11 ; till 1 coincides with 6, then 2 will coincide with 12; and thus continue to do till you have gone through the line, and if the Rule is correctly graduated, each number will correspond as above stated ; if they do not correspond, the Rule is not correct, and consequently will aot give a correct answer. 13 GUNTER’S SCALE. Gunter's Scale has upon it eight lines : 1. Sine Rhombs, (marked SR) corresponding to the Logarithms of Natural Sines of every point of the Mariner's Compass, numbered from the left hand towards the right, with 1, 2, 3, 4, 5, 6, 7 to 8, where is a brass pin ; this line is also divided, where it can be done, into halves and quarters. 2. Tangent Rhombs, (marked TR) corresponds to the Logarithms of the tangents of every point of the Compass, and is numbered 1, 2, 3 to 4, at the right hand, where there is a pin ; and thence towards the left hand with 5, 6, 7 ; it is also divided, where it can be done, into halves and quarters. 3. The line of numbers, (marked NUM) corresponds to the Logarithms of numbers, and is marked thus : at the left hand it begins at 1, and towards the right hand are 2, 3, 4, 5, 6, 7, 8, 9, and 1 in the middle ; at which, is a brass pin ; then 2, 3, 4, 5, 6, 7, 8, 9 and 10 at the end, where there is another pin. The value of these numbers and their intermediate division, depends on the estimated values of the extreme numbers, 1 and 10 ; and as this line is of great importance, a particular description of it and its uses, will be given. The first 1, may be counted for 1, or 10, or 100, or 1000, and then the next 2 is accordingly, 2, or 20, or 200, or 2000, &c. Again, the first 1 may be reckoned one tenth or one hundredth, or one thousandth part, &c. and then the next 2 is two tenth, or two hundredth, or two thousandth parts, &c. &c. Then, if the first 1, be reckoned 1, the middle 1 is reckoned 10 and 2, at its right hand is 20, 3 is 30, 4 is 40, and 10, at the end is 100; Again, if the first 1 is 10, the next 2 is 20, 3 is 30, and so on, making the middle 1, 100, the next 2 is 200, the next 3 is 300, 4 is 400, and 10 at the end is 1000. In like manner, if the first 1 be es. teemed one tenth part, the next 2 is two tenth parts, and the middle 1 is one, and the next 2 is two, and 10 at the end is ten. Again, if the first 1 be counted one hundredth part, the next 2 is two hundredth parts, the middled is now ten hundredth parts, the next two hundredth parts, the B |